PSTAT 120B Quiz 1
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Quiz 1
PSTAT 120B
Instructions: the quiz is open book and open note and has no strict time limit. You can use any course materials (and are encouraged to do so!). Complete the quiz on your own and do not share your work with anyone else until after the deadline has passed. Write your answers on a printed copy of the quiz or on blank paper and upload a PDF copy to Gradescope by the deadline. Please be sure to match the pages of your PDF to the outline.
1. Let Y1 , Y2 uniform(0, 1), and consider U = Y12 + Y22 . In the following parts you will find the distribution of U .
(a) The plots below show the support set of (Y1 , Y2 ).
Draw the boundary U = r2 for some r e (0, 1] and shade the region {U < r2 }.
1
1
1 y2
Draw the boundary U = r2 for some r e ╱ 1, ^2、and shade the region {U < r2 }.
1
1
1 y2
(b) Find P (U < r2 ) for r e (0, 1] by calculating the area of the shaded region in the left
plot.
(c) Find P (U < r2 ) for r e (1, ^2) by calculating the area of the shaded region in the right plot. (Hint: write the area as the sum of three subregions – a circular section and two equally-sized triangles.)
(d) Write the CDF of U piecewise by replacing r2 with u in the above expressions. Check that your answer is in fact a CDF.
2. Let Y1 , . . . , Yn − 1 χ1(2) and consider U = Yi . (a) Show that U ~ χn(2) − 1 . (Hint: use the MGF method.) (b) What is the expectation of U?
(c) If σ 2 is a constant, what is the expectation of U?
2022-07-27